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math::combinatorics(n)                        Tcl Math Library                        math::combinatorics(n)



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NAME
       math::combinatorics - Combinatorial functions in the Tcl Math Library

SYNOPSIS
       package require Tcl  8.2

       package require math  ?1.2.3?

       ::math::ln_Gamma z

       ::math::factorial x

       ::math::choose n k

       ::math::Beta z w

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DESCRIPTION
       The math package contains implementations of several functions useful in combinatorial problems.

COMMANDS
       ::math::ln_Gamma z
              Returns the natural logarithm of the Gamma function for the argument z.

              The Gamma function is defined as the improper integral from zero to positive infinity of

                t**(x-1)*exp(-t) dt


              The  approximation  used in the Tcl Math Library is from Lanczos, ISIAM J. Numerical Analysis,
              series B, volume 1, p. 86.  For "x > 1", the absolute error of the result  is  claimed  to  be
              smaller than 5.5*10**-10 -- that is, the resulting value of Gamma when

                exp( ln_Gamma( x) )

              is computed is expected to be precise to better than nine significant figures.

       ::math::factorial x
              Returns the factorial of the argument x.

              For integer x, 0 <= x <= 12, an exact integer result is returned.

              For integer x, 13 <= x <= 21, an exact floating-point result is returned on machines with IEEE
              floating point.

              For integer x, 22 <= x <= 170, the result is exact to 1 ULP.

              For  real  x,  x  >=  0,  the  result  is  approximated  by  computing  Gamma(x+1)  using  the
              ::math::ln_Gamma  function,  and the result is expected to be precise to better than nine sig-nificant significant
              nificant figures.

              It is an error to present x <= -1 or x > 170, or a value of x that is not numeric.

       ::math::choose n k
              Returns the binomial coefficient C(n, k)

                 C(n,k) = n! / k! (n-k)!

              If both parameters are integers and the result fits in 32 bits, the result is  rounded  to  an
              integer.

              Integer results are exact up to at least n = 34.  Floating point results are precise to better
              than nine significant figures.

       ::math::Beta z w
              Returns the Beta function of the parameters z and w.

                 Beta(z,w) = Beta(w,z) = Gamma(z) * Gamma(w) / Gamma(z+w)

              Results are returned as a floating point number precise to better than nine significant digits
              provided that w and z are both at least 1.


BUGS, IDEAS, FEEDBACK
       This  document,  and  the  package  it  describes,  will undoubtedly contain bugs and other problems.
       Please  report  such  in  the   category   math   of   the   Tcllib   SF   Trackers   [http://source -
       forge.net/tracker/? group_id=12883].   Please  also report any ideas for enhancements you may have for
       either package and/or documentation.

CATEGORY
       Mathematics



math                                                1.2.3                             math::combinatorics(n)